The Gold Standard of Hydrology | Partial-Duration Series, Annual Maxima and Independent Flood Peaks

Full chapter index

An annual maximum series, or AMS, contains the largest instantaneous peak in each year of record. A 30-year gauge record normally contributes 30 values, including years whose largest flow is fairly ordinary. The sampling rule is easy to audit, and annual maxima are the basis of widely used procedures such as the United States federal Bulletin 17C framework.

A partial-duration series, or PDS, contains peaks above a selected base discharge. It is also called a peaks-over-threshold series. A wet year can contribute several peaks; a dry year can contribute none. When the goal is frequent-event estimation, this can use relevant information that the annual maximum rule discards. The gain is conditional on selecting distinct events and treating the exceedance rate correctly.

The two series should not be mixed casually. Rank 30 annual maxima and rank 45 threshold exceedances from the same period, and the second ranks are not simply “more years.” Their plotting positions and probability interpretations differ. An AMS recurrence interval refers to annual maxima. A PDS model combines the magnitude distribution above the threshold with the average occurrence rate of independent events.

Choose the sampling frame from the decision. A bridge design standard may mandate an annual-maximum method. An operations team may care about flows exceeded several times per decade. A geomorphology study may define events by sediment-moving discharge rather than by a statistical threshold. State the target, record and governing guidance before choosing the series.

Contents · Next chapter

A table of peak dates and flows hides the shape between peaks. Plot the continuous discharge record, missing intervals, rating changes and any quality flags. A double crest separated by six hours may be one storm response. Two equal peaks separated by a return to near-baseflow and a different rainfall system may be independent events. The hydrograph carries that context.

Event identification needs a time rule and often a recession rule. A time rule might require peaks to be separated by at least several days. A recession rule might require discharge to fall below a fraction of the smaller peak, or below the threshold, before another peak can count. The suitable values depend on catchment response time, storage and season. A single universal gap is not defensible for every basin.

Overlay precipitation, snowmelt indicators, reservoir operations and tide where relevant. These variables do not automatically decide independence, but they help explain why peaks cluster. In a small tropical catchment, separate convective storms can occur within a day. In a large snowmelt basin, a week of high flow may be one sustained event with weather-driven pulses.

Mark missing data before extracting peaks. A threshold method can silently miss events in a gap, while an annual series may still retain a maximum from another part of the year. Neither is automatically safe. Document whether a year is complete enough for AMS and how exposure time is calculated for PDS.

Contents · Previous chapter · Next chapter

A low threshold yields many exceedances but risks including ordinary flows, dependent shoulders and a magnitude distribution that does not behave like the tail model. A high threshold isolates the tail but leaves few events and wide uncertainty. Threshold choice therefore balances model bias against sampling variance.

Begin with physical and statistical views. Plot the number of exceedances against candidate thresholds, inspect mean excess or parameter stability diagnostics where appropriate, and view the selected peaks on the hydrograph. A stable-looking diagnostic is not sufficient if it counts multiple crests from one storm. A physically tidy catalogue is not sufficient if the chosen threshold changes results drastically.

Do not choose the threshold because it produces exactly one event per year on average. That convention can be a useful starting point, but independence and tail behaviour still need checking. In some basins the defensible rate is lower; in others a lower threshold with rigorous declustering may retain more independent events.

Record the threshold units, datum and any transformation. Rating-curve revisions can change historical discharge estimates. If the threshold is 500 cubic metres per second under one rating and the record is later reprocessed, re-extract the events. A PDS catalogue is derived data tied to the underlying series, not a timeless list.

Contents · Previous chapter · Next chapter

Suppose a five-year record has candidate peaks above 120 cubic metres per second. In Year 1, flows reach 142 on 3 January and 136 on 4 January without falling below 110 between them. In Year 2, peaks of 151 and 129 occur six weeks apart with full recession. Year 3 has no exceedance. Year 4 has peaks of 188, 173 and 131; the first two belong to one three-day storm, while the third follows a separate storm two months later. Year 5 has a single peak of 160.

With a rule requiring a return below 110 and separate storm evidence, the catalogue keeps one event from Year 1, two from Year 2, none from Year 3, two from Year 4 and one from Year 5. The retained rate is six events over five complete record-years, or 1.2 events per year. The larger crest represents each clustered episode.

The AMS from the same record is 142, 151, the below-threshold annual maximum for Year 3, 188 and 160. It contains five values. Notice what each rule does: AMS keeps the dry-year maximum and discards the second independent Year 2 event and the later Year 4 event. PDS keeps those independent exceedances and omits Year 3 because no event crossed 120.

Now change the recession criterion from 110 to 100. If the Year 2 hydrograph never fell below 100 between its two peaks, they merge and the estimated event rate becomes 1.0 per year. The data did not change; the event definition did. That sensitivity should be reported and can be more important than a minor change in fitted distribution.

Contents · Previous chapter · Next chapter

Statistical formulas often assume event magnitudes or occurrences are independent. Independence is not created by deleting adjacent rows. It is a claim that, after the chosen event rule and conditioning, one retained peak does not carry material information about another beyond the model.

Storm separation, return to baseflow and catchment travel time are evidence for that claim. The existing guide to lag time and catchment clocks helps distinguish response measures that inform a declustering window. Antecedent wetness can connect otherwise separate storms; use the antecedent precipitation index guide to see why catchment memory persists after rain stops.

Check clustering by season and storm system. If every monsoon spell contributes several peaks, a fixed 24-hour gap may under-decluster. If isolated convective storms dominate, a long gap may merge genuinely distinct events. Compare candidate catalogues and document which peaks split or merge.

Independence also applies to the record over time. Land-use change, river training, reservoir construction or gauge relocation can change the generating process. Declustering does not repair nonstationarity. Split-period plots, metadata and physical explanation are needed before treating the full record as one homogeneous sample.

Contents · Previous chapter · Next chapter

A PDS analysis needs both exceedance magnitude and occurrence rate. Under a simple Poisson occurrence model with mean rate lambda independent events per year, if the probability that an event exceeds a selected magnitude is q, the expected rate above that magnitude is lambda q. The probability of at least one such event in a year is 1 minus exp(−lambda q).

This formula explains why a PDS “return period” cannot be read as the rank of annual maxima. An event-level average recurrence interval of 1/(lambda q) is not numerically identical to the annual exceedance probability except when the probability is small. For common events, the difference matters.

Use a fictional example. Let lambda be 1.5 events per year, and suppose 20 per cent of threshold events exceed 300 cubic metres per second. The rate above 300 is 0.3 per year. Under the Poisson model, annual exceedance probability is 1 − exp(−0.3), about 0.259, not 0.300. The corresponding annual-probability return period is about 3.86 years.

The model assumptions need checking. Event rates may vary by season or climate regime, and counts can be overdispersed relative to Poisson. If occurrence and magnitude are dependent, a simple product is inadequate. Report the transformation used and avoid presenting a rounded “100-year flood” without its annual exceedance probability and uncertainty.

Contents · Previous chapter · Next chapter

The attached fictional table compares estimates expressed as annual exceedance probabilities. This common scale is essential. A PDS quantile labelled by event recurrence and an AMS quantile labelled by annual probability can otherwise appear to disagree when the labels differ more than the fitted floods.

At rare probabilities and under compatible assumptions, AMS and PDS estimates may converge because the largest events dominate both. At more frequent probabilities, PDS can use additional independent floods and often provides a more detailed view. This is not a guarantee of lower error. Threshold, declustering and rate uncertainty enter the result.

Compare fitted curves with empirical points and the raw hydrograph. If one method predicts a frequent quantile below the extraction threshold, something is inconsistent. If a rare PDS estimate is controlled by one enormous event, the nominal event count may exaggerate precision. Plot confidence or credible intervals and identify influential events.

Use the same record period, quality control and treatment of historical information when comparing methods. An AMS fit that includes paleoflood bounds or historical peaks is not directly comparable to a PDS fit using only systematic gauge years. Method labels do not replace a data provenance table.

Contents · Previous chapter · Next chapter

AMS usually requires a decision about whether each year has sufficient coverage to identify its maximum. PDS requires the amount of time during which an exceedance could have been observed. A gap during the flood season is more consequential than the same duration during a reliably dry season, but deleting exposure selectively can also bias the rate.

Create a record-availability calendar. For every year, mark continuous coverage, partial coverage, rating limits and periods when flows exceeded measurable range. If a major flood is known only to exceed an instrument limit, treat it as censored information rather than replacing it with the limit value.

Historical accounts can extend knowledge of large floods, but the observation threshold may change over time. A town chronicle might record only damaging events, while a modern gauge captures all flows above the statistical threshold. Combine these data only with a model that represents perception and censoring; otherwise the apparent past event rate will be too low.

The transfer check is to imagine moving the same numerical series to a different gauge history. If the series came from an unchanged, continuously operating gauge, the catalogue may be straightforward. If it came from merged stations with different ratings, the same peaks require additional uncertainty. Numbers do not carry their measurement conditions automatically.

Contents · Previous chapter · Next chapter

Extreme-value models are extrapolation tools, not machines that turn thirty values into certainty. Inspect whether the fitted tail is compatible with the retained events, whether parameters change across reasonable thresholds, and whether residual or probability plots reveal systematic curvature. Diagnostics should include the event-selection sensitivity, not only the final magnitude fit.

Bootstrap or other interval procedures must respect the sampling structure. Resampling individual peaks as though they were independent is only defensible after declustering and may still ignore uncertainty in lambda or threshold choice. Block or year-based resampling can preserve some structure, depending on the model.

Parameter uncertainty is not the only uncertainty. Rating curves at extreme stage, basin change, missing peaks, historical thresholds and model form can dominate. Separate these where possible. A narrow mathematical interval conditional on a disputed rating curve is not a comprehensive uncertainty statement.

Use leave-one-event-out checks to identify leverage. If removing one peak changes a rare quantile dramatically, report that dependence. The result may still be the best available estimate, but decision-makers need to know whether the curve is supported by many events or anchored by one observation.

Contents · Previous chapter · Next chapter

“Partial duration means every local maximum” is false. It means every independent event above a threshold under a declared rule. “Annual maximum wastes data, so PDS is always better” is also false. AMS has a simple sampling frame and strong standardised methods; PDS adds choices that can improve or weaken the analysis.

“A ten-year flood happens once every ten years” misstates probability. A 0.10 annual exceedance probability can occur in consecutive years or not occur for decades. For a PDS, an event recurrence label may refer to average occurrence rate rather than annual probability. Always name the probability scale.

“More events mean narrower uncertainty” is incomplete. Dependent events, a low threshold or changing observation conditions do not add independent tail information in proportion to the row count. Effective information matters more than catalogue length.

Finally, frequency analysis does not predict the date of the next flood and does not by itself establish consequences. Hydraulic modelling, exposure, vulnerability, debris, coincident tide and failure modes may determine risk. Keep hydrologic frequency separate from the broader risk assessment while linking them through explicit probabilities.

Contents · Previous chapter · Next chapter

Practice one: a year contains peaks of 210, 195 and 205, each two days apart, with discharge never below 170. Under a threshold of 180, the raw list has three exceedances. Before fitting, inspect rainfall and recession. A single slow storm should contribute one event, commonly represented by 210, not three independent floods.

Practice two: raising the threshold from 180 to 220 reduces the catalogue from 42 to 19 events and changes lambda from 1.4 to 0.63. The fitted rare quantile barely moves, but a two-year event changes substantially. That pattern can be reasonable: the higher threshold focuses the tail while frequent estimates depend strongly on the discarded range. Report both sensitivity and intended use.

Practice three: the same basin record is divided by a reservoir commissioning date. Post-reservoir peaks are lower, and releases create secondary crests. A single stationary PDS is hard to defend. Analyse regimes separately or model the operational change explicitly. Do not treat declustering as a substitute for a causal break.

Transfer the method to heatwaves, pollution episodes or service outages. Threshold exceedances are useful only after episodes are defined. Closely spaced readings can belong to one event, observation coverage affects the rate, and event probability differs from annual probability. The hydrologic lesson is a general lesson in counting extremes without counting the same cause twice.

Contents · Previous chapter · Next chapter

Which series should I use?

Use the method required by the governing standard and suited to the decision. AMS is simple and standardised for many design applications. PDS is useful when multiple independent threshold events carry relevant information and the additional assumptions can be supported.

How do I choose the independence gap?

Use basin response, hydrograph recession, storm evidence and sensitivity analysis. A fixed number of days without hydrologic justification is only a convention.

Can a year have no PDS event?

Yes. If no peak crosses the threshold, that year contributes exposure time but no event. This is different from a missing year.

Are AMS and PDS return periods interchangeable?

No. Translate results to a common annual exceedance probability and state the occurrence model before comparing them.

Authoritative reading includes the USGS Bulletin 17C guidelines, which describes annual maximum series and notes partial-duration analysis, and the USGS introduction and methods for peak streamflow analysis, which defines a partial-duration series as flows above a selected base regardless of the number per year. Use the current local or national flood-frequency guidance for an actual design.

Return to The Gold Standard of Hydrology for the subject map. Read flow-duration curves and the drought sequences they hide when the question concerns the whole flow distribution rather than flood events, and paired catchments and attribution of change when a changing basin, not only a changing frequency curve, is the scientific problem. The next useful move is to plot the hydrograph, label candidate episodes, and make every merge or split reproducible before fitting anything.

Contents · Previous chapter · Return to the Hydrology guide